Capacity of Erasure Networks in the Presence of Spatial Network Coding

We study the capacity of a point-to-point erasure network in the presence of a restricted form of network coding, to which we refer as spatial network coding. In spatial network coding, the temporal aspect of coding is absent, i.e., a coding relationship is not created among successive packets sent on each network link, in order to combat link erasures. For a given source-destination pair, there exists a minimum cut rate corresponding to each random instance of the network We prove that the statistical average of such minimum cut rates equals the network capacity for the corresponding unicast session. We also show that this capacity can be higher than the capacity achieved through pure routing. This stands in contrast to the known result for deterministic networks where, in the case of a unicast session, capacities achieved through routing and network coding are equal. In the case of a multicast session, we show that, similar to deterministic networks, the capacity achievable through spatial network coding is equal to the minimum of the capacities of the constituent unicast sessions.